Graph Theory And Bioinformatics Jason Wengert

Outline Introduction to Graphs Eulerian Paths & Hamiltonian Cycles Interval Graph & Shape of Genes Sequencing DNA Shortest Superstring Problem Sequencing by Hybridization (SBH) SBH as Hamiltonian Cycle Problem SBH as Eulerian Path Problem

Graph Theory A graph is a set of vertices connected by edges Edges connect two vertices and can be weighted, directed or undirected

The Bridge Obsession Problem Bridges of Königsberg Find a tour crossing every bridge just once Leonhard Euler, 1735

Eulerian Cycle Problem Find a cycle that visits every edge exactly once Linear time Solutions exist if each vertex has even degree For a path, the start and end vertices can have odd degree More complicated Königsberg

Hamiltonian Cycle Problem Find a cycle that visits every vertex exactly once NP – complete  Equivalent to Travelling Salesperson Problem Game invented by Sir William Hamilton in 1857

Interval Graph Vertices are segments on a line Edges exist between two vertices if any part of their segments overlap

Seymour Benzer and T4 phages Benzer studied mutated bacteriophages Non-mutated T4 phages kill bacteria The mutants had continuous segments missing from their DNA These mutants lost the ability to kill the bacteria Two mutant strains with non-overlapping segments missing can kill bacteria together

Complementation between pairs of mutant T4 bacteriophages

Interval Graph: Linear Genes

Interval Graph: Branched Genes

Sequencing DNA Sanger Method:  Make copies of fragments of different length by nucleotide starvation during copying  Separate the fragments via electrophoresis  Each starvation experiment produces a ladder of fragments of various lengths  Generate several-hundred-nucleotide fragment sequences (reads) from these ladders

Sanger Method: Generating Read 1. Start at primer (restriction site) 2. Grow DNA chain 3. Include ddNTPs 4. Stops reaction at all possible points 5. Separate products by length, using gel electrophoresis

Combining Reads Need a method to take the reads and combine them into a genome You don't know where the reads came from in the DNA

Shortest Superstring Problem Problem: Given a set of strings, find a shortest string that contains all of them Input: Strings s 1, s 2,…., s n Output: A string s that contains all strings s 1, s 2,…., s n as substrings, such that the length of s is minimized Complexity: NP – complete Note: this formulation does not take into account sequencing errors

Shortest Superstring Problem: Example

Reducing SSP to TSP Define overlap ( s i, s j ) as the length of the longest prefix of s j that matches a suffix of s i. aaaggcatcaaatctaaaggcatcaaa Construct a graph with n vertices representing the n strings s 1, s 2,…., s n. Insert edges of length overlap ( s i, s j ) between vertices s i and s j. Find the shortest path which visits every vertex exactly once. This is the Traveling Salesman Problem (TSP), which is also NP – complete.

Sequencing By Hybridization The presence of a target DNA sequence can be determined by creating a probe, a single-strand fragment with the complementary sequence If the target sequence exists in the sample, it will hybridize with the probe

How SBH Works Attach all possible DNA probes of length l to a flat surface, each probe at a distinct and known location. This set of probes is called the DNA array. Apply a solution containing fluorescently labeled DNA fragment to the array. The DNA fragment hybridizes with those probes that are complementary to substrings of length l of the fragment.

How SBH Works (cont’d) Using a spectroscopic detector, determine which probes hybridize to the DNA fragment to obtain the l–mer composition of the target DNA fragment. Apply the combinatorial algorithm (below) to reconstruct the sequence of the target DNA fragment from the l – mer composition.

Hybridization on DNA Array

Spectrum Spectrum(s,l) is the multiset of all l-mers that appear in s Example:  Spectrum(TATGGTGC, 3) = {TAT, ATG, TGG, GGT, GTG, TGC}

The SBH Problem Goal: Reconstruct a string from its l-mer composition Input: A set S, representing all l-mers from an (unknown) string s Output: String s such that Spectrum ( s,l ) = S

SBH As Hamiltonian Path Vertices of the graph are every l-mer in the spectrum A directed edge leads from vertex u to vertex v if the last l-1 characters of u equal the first l-1 characters of v. S = { ATG TGG TGC GTG GGC GCA GCG CGT }

SBH: Hamiltonian Path Approach S = { ATG TGG TGC GTG GGC GCA GCG CGT } Path 1: ATGCGTGGCA ATGGCGTGCA Path 2:

SBH As Eulerian Path Vertices are (l-1)-mers Edges represent the l-mers from the spectrum AT GT CG CA GC TG GG S = { ATG, TGC, GTG, GGC, GCA, GCG, CGT }

SBH: Eulerian Path Approach S = { AT, TG, GC, GG, GT, CA, CG } corresponds to two different paths: ATGGCGTGCA ATGCGTGGCA AT TG GC CA GG GT CG AT GT CG CA GCTG GG

Euler Theorem A graph is balanced if for every vertex the number of incoming edges equals to the number of outgoing edges: in(v)=out(v) Theorem: A connected graph is Eulerian if and only if each of its vertices is balanced.

Constructing an Eulerian Cycle Start with an arbitrary vertex and form an arbitrary cycle until you reach a dead end  Because the graph is Eulerian, this dead end will be the starting point If the resulting cycle is not Eulerian, select a vertex from the cycle with untraversed edges, and repeat step 1 starting there. Combine the two cycles and repeat step 2.

Constructing an Eulerian Cycle

Some Difficulties with SBH Fidelity of Hybridization: difficult to detect differences between probes hybridized with perfect matches and 1 or 2 mismatches Array Size: Effect of low fidelity can be decreased with longer l-mers, but array size increases exponentially in l. Array size is limited with current technology. Practicality: SBH is still impractical. As DNA microarray technology improves, SBH may become practical in the future Practicality again: Although SBH is still impractical, it spearheaded expression analysis and SNP analysis techniques

Summary Introduction to Graphs Eulerian Paths & Hamiltonian Cycles Interval Graph & Shape of Genes Sequencing DNA Shortest Superstring Problem Sequencing by Hybridization (SBH) SBH as Hamiltonian Cycle Problem SBH as Eulerian Path Problem

References Jones, N. and Pevzner, P. An introduction to Bioinformatics Algorithms. Some slides courtesy of the book's website at bix.ucsd.edu/bioalgorithms/slides.php